On the classification of two-qubit group orbits and the use
of coarse-grained ‘shape’ as a superselection property
Thomas Hebdige1 and David Jennings1,2,3 1
Controlled Quantum Dynamics Theory Group, Imperial College London, Prince Consort Road, London SW7 2BW, UK 2
Department of Physics, University of Oxford, Oxford, OX1 3PU, UK 3
School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, UK. January 30, 2019
Recently a complete set of entropic con-ditions has been derived for the intercon-version structure of states under quan-tum operations that respect a specified symmetry action, however the core struc-ture of these conditions is still only par-tially understood. Here we develop a coarse-grained description with the aim of shedding light on both the structure and the complexity of this general prob-lem. Specifically, we consider the degree to which one can associate a basic ‘shape’ property to a quantum state or channel that captures coarse-grained data either for state interconversion or for the use of a state within a simulation protocol. We provide a complete solution for the two-qubit case under the rotation group, give analysis for the more general case and dis-cuss possible extensions of the approach.
1
Introduction
Symmetry principles are ubiquitous in both the classical and quantum realms. They constrain dynamics, connect with conservation laws and simplify computations of physical properties. In crystallography the crystal structure is described by a symmetry group and largely determines the properties of the material [1].
Symmetry considerations have been extremely successful in the field of quantum information theory [2–4]. Moreover a substantial component of this work has addressed the regime in which symmetry principles disconnect from conserva-tion laws [5], and which has found application in the study of quantum features of thermodynam-ics [6–8]. Quantum states that break the symme-try can also be used to circumvent limitations on
Figure 1: Residual symmetry-types in two-qubit sys-tems. We associate to each state a ‘shape’ that de-scribes the residual symmetries of that state. As an ex-ample, in the above figure we show the symmetry proper-ties of the set of all mixtures of Bell states underSU(2). This is determined solely by the triplet component of the state. The only possibilities are for the state to have a residual symmetry of (a) a cylinder, (b) a double-sided rectangle, or (c) a sphere. This ‘shape’ can be used to provide superselection rules when using the two-qubit state to simulate quantum operations. Details are given in section 4.
the precision of measurements imposed by conser-vation laws, as described by the WAY theorem [9–14], while metrology can be viewed as using broken symmetries to distinguish different group transformations [15]. Symmetry principles have also provided new insights into quantum speed limits and how quickly quantum operations can be performed [16]. Finally, symmetry groups pro-vide an invaluable toolkit in the context of quan-tum computing [17–20].
In this work we look at how general quantum states ρ can be classified by groups that break a
given symmetry constraint to an equal degree. A concrete example makes this clearer: both a chair and an arrow break rotational symmetry, however an arrow has a residual (axial) symmetry group to it, and so breaks rotations to a weaker degree than the chair. For a fixed quantum system of dimensiond, and symmetry actionG, we can ask what residual symmetries does the system admit, and how do these residual symmetries transform under symmetric dynamics. We also address the degree to which this aspect of a quantum system constitutes a property in the ‘resource-theoretic’ sense.
The structure of the paper is as follows. In the next section we provide notation and back-ground motivation. In section 3 we provide the coarse-graining scheme and some basic relations involved. This demonstrates the complexity of the topic, and highlights some subtleties if one wishes to associate the scheme to a quantum re-source. In section4we provide a complete classi-fication for two-qubit systems (partly pictured in Fig. 1and more fully in Fig. 5), and which illus-trates how quantum state spaces admit a coarse-grained partial order.
2
Background and Motivation
In this paper we shall use the following nota-tion. For any quantum system A we denote by HA its associated Hilbert space, and B(HA) the set of linear operators on HA. We assume that a symmetry action of G is defined on HA through the unitary representation g 7→ U(g) ∈ B(HA). The group action on a quantum state ρ∈ B(HA)is given by the adjoint actionUg(ρ) :=
U(g) ρ U†(g).
For any quantum channel E : B(HA) → B(HA0), the action of the group on E isUg(E) := Ug◦ E ◦ Ug−1 (where◦denotes the concatenation
of channels). This can be interpreted as moving to a different frame, performing the channel and returning to the original frame.
If Ug(E) = E for all g ∈ G then E is called a symmetric operation. This includes certain preparation procedures, and the resulting states are known as symmetric states because they are invariant under the group action: Ug(ρ) = ρ for all g∈G.
2.1 General state transformations under a symmetry constraint
A primary concern here is the study of general quantum states or quantum operations under a symmetry action. In particular we would like to study the degree to which any given quan-tum state breaks the symmetry (a ‘resource state’ [21, 22]). One way of comparing states in this manner is to say that ρ σ if and only if ρ → σ =E(ρ) for some symmetric quantum op-eration E, which defines a partial order on quan-tum states [23, 24]. A measure of how much a state breaks the symmetry is then a function M that respects the partial order – namely, if ρσ then M(ρ) ≥ M(σ) [21, 22]. A central ques-tion is whether a complete set of measures ex-ist that fully capture this ordering of states, and thus asks when two quantum states can or can-not be interconverted under the symmetry. Very recently [7] an answer was given for this general problem in terms of single-shot entropies of the form Hmin(R|A) that quantify the amount of in-formation in a systemAabout an external quan-tum reference frame R. Thus it is the case that ρ→σ under the symmetry constraint if and only if Hmin(R|A) decreases with respect to all refer-ence frames R.
However this set of measures for the resource theory is over-complete and highly complex to use. Applying it to general systems requires so-phisticated techniques in single-shot information theory. Here we wish to tackle a more mod-est goal, and instead of describing the complete symmetry properties of a quantum state, wish to coarse-grain states into groups that break the symmetry in the same way. Notably, this itself turns out to be a highly non-trivial problem and so sheds light on the structure of the more general set of measures given in [7].
symmetric dynamics therefore constitutes a mea-sure of its properties.
In [25], the symmetry properties of quantum channels were analysed, with a focus on the orbit of a channelE, defined as
M(E) :={Ug(E) : g∈G}. (1)
A similar definition applies to the orbit of states under the group, so M(ρ) ={Ug(ρ) : g∈G}.
Now the type of orbit one obtains provides a natural context in which to understand both how the channel breaks a symmetry constraint, and moreover how one might simulate such a channel using an auxiliary quantum system B prepared in a non-symmetric stateσB.
It is clear that an auxiliary resource state may be used with varying efficiency in the simulation of a channel on a system S, however in [25] the question of when a state σB can be used to in-duce a channelE such that its ability to simulate the very same channel on subsequent systems is undiminished. This was motivated by the dis-covery of a “catalytic coherence” protocol [26] in which a quantum stateσB is used as a phase ref-erence. The state undergoes non-trivial changes σB → σB(1) → σ
(2)
b → · · ·σ (n)
B yet for all n ≥ 1 its abilities as a phase reference never diminish. In [25] the structure of sucharbitrarily repeatable
protocols was spelled out using harmonic anal-ysis. It was shown that if a channel E on S is simulated using a quantum stateσBunder an ar-bitrarily repeatable protocol then there exists a POVM measurement {Mk}on B such that
E(ρ) =X
k
tr(Mkσ)Φk(ρ) (2)
where{Φk} are completely positive maps on the primary system S. The POVM {Mk} on the auxiliary system depends explicitly on the tar-get channel E. Crucially however, it can be un-derstood as resolving the location x of E on its unitary orbit M(E) to some scale that depends on the dimensiondofS. For example, in the case of catalytic coherence, the POVM measurement on B corresponds simply to resolving a point x
on a circle (the orbit of the channelEunder phase rotations) down to an angular scale∼ 2πd.
The degree to which x is resolved on M(E)
clearly depends on the resource state σB, and corresponds to the fact that σB is a quantum reference frame. This in turn can be viewed as
inducing non-classical geometry on M(E) – for example, for G = SU(2) a channel can have orbit M(E) that is a 2-sphere. The use of a bounded reference system in this case means that only finite resolution of points on M(E) is pos-sible, however since the 2-sphere carries a phase space structure the bounded reference case cor-responds to the so-called “fuzzy sphere” model from non-commutative geometry [27]. This dif-fers from catalytic coherence in that we now have complementarity in resolving the coordinates on the sphere, and which ultimately arises because the two orbits have quite different structure.
Thus the general efficient use of a quantum state σB to simulate a channel E on S can be analysed in terms of two aspects:
1. (‘Shape’) The type of orbit M(σ) the state has under the group action compared to M(E), independent of metrical aspects.
2. (‘Geometry’) The ability ofσto encode clas-sical coordinate data forx inM(E).
What is a necessary relation between M(E) and M(σ)? It is clear that we need σ to break the symmetry to a ‘larger’ degree than E, however in this paper we would like to make this statement more precise. Given the complexity of the general problem, one motivation in this work is to start with the above separation into ‘shape’ and ‘ge-ometry’, and develop an organisational setting in which we provide a coarse-graining over states of the same ‘shape’, with the remaining task being to analyse the ‘geometry’ aspect of the state.
2.3 Related topics in quantum state tomogra-phy and quantum computation
how they transform among themselves is of po-tential relevance to quantum tomography under symmetry constraints.
A wholly practical direction where such analy-sis might be of relevance is in the computational power of gate-sets in quantum computation, and how such gate-sets interact with states that in-crease the computational power of the gate-set (such as magic states for the stabilizers [29]). For example, a recent work provides a classification of all Clifford gate-sets [17] in a lattice hierar-chy, and so aspects of the present work might be applicable in studying how noisy quantum states increase the computational power of easily realis-able gate-sets.
2.4 Core questions of the present work
In subsection2.1we highlighted that state inter-conversion ρ → σ under a symmetry constraint is highly non-trivial, and while progress has been made on this fundamental problem, it is not clear how complex it is in general. Also, as discussed in subsection2.2recent results on optimal proto-cols show that the structure of this theory natu-rally decomposes into a geometric component and the particular kind of group orbit the quantum channel has. In light of these considerations one expects that the group orbit level of description might provide a form of coarse-grained descrip-tion that sheds light on the more complex parent resource theory. By studying this coarse-grained theory we can shed light on the parent theory, and moreover extend the notion of a resource theory from being about pre-orders on quantum states to considering pre-orders onsub-setsof states. How-ever, in order for this to work and be meaningful, it is crucial that the coarse-grained theory be (a) physically sensible and (b) relate naturally to the parent theory. Therefore the core questions we address in this work are the following:
1. Does a coarse-grained resource-theory frame-work exist?
2. Is this theory consistent with its parent re-source theory? Is it physically sensible?
3. How complex is the general structure of symmetry-based resource theories?
In the next section we tackle the first two of these questions, by showing how a natural par-tition of the state space arises that is consistent
with the finer-grained resource theory, and de-scribe its limitations and physical robustness. In the process, and through exhaustive analysis of theG=SU(2)case for two qubits, we shed light on the crucial third question posed.
3
Coarse-graining states and channels
under a Symmetry Action
3.1 Simulating Operations Under Symmetry Constraints
If one wishes to realise a quantum channel E via a symmetric interaction with a state σ, the most natural way to model this is as
E(ρ) = trBV(ρA⊗σB)V† (3)
where V is a covariant unitary, namely
[V, UA(g)⊗UB(g)] = 0 for all g ∈ G. If such a relation holds, we say that E can be simulated
using the state σB.
A basic classification of states and channels under a symmetry can be done by studying the subgroup of residual symmetries for the state or channel, called the isotropy subgroup or stabilizer of E and defined as
Iso(E) :={g∈G : Ug(E) =E}. (4)
It specifies the residual symmetry that the nel has under the group action. Symmetric chan-nels are invariant under all group transforma-tions, and simply have Iso(E) = G. This notion also applies to states, where Iso(ρ) := {g ∈ G :
Ug(ρ) =ρ}.
In many cases the quantum operation E has some residual symmetry, which is characterised an isotropy subgroup H of G. The possible isotropy subgroups form an abstract structure called a lattice [30]. The partial ordering of the subgroups is defined by subset inclusion, namely H1 ≺H2ifH1 ⊆H2forH1, H2 ⊆G. In addition, every pair of elements have a unique supremum and unique infimum, which define binary opera-tions called the meet and join [31]. For two sub-groupsH1andH2, their meet is denotedH1∧H2 and defined asH1∩H2, while their join is denoted H1 ∨H2, defined as the subgroup generated by H1 ∪H2 [30]. This structure is illustrated by a Hasse diagram of the subgroups, as shown in Fig.
Figure 2: Coarse-grained classification of quantum states under a symmetry. The set of all density operators is partitioned into setsC(H)with definite symmetries (seen on the left). The C(H) correspond to the subgroups shown in the Hasse diagram in the middle. The Hasse diagram represents the subgroup lattice, with lines indicating subgroup inclusion, i.e. H5 ≺ H4 ≺ H3 etc. Note that for finite-dimensional systems many of theC(H)will be
empty. The subsetsC(H)can be associated, up to diffeomorphisms, with group orbits (indicated on the right). The left figure shows the action of group-averaging (overH2), going fromσtoPH2(σ)as described in section3.4.
Since quantum operations can be combined in a number of ways, we present some basic state-ments that constrain the residual symmetry of the resultant quantum operation. The proofs for this section are provided in Appendix A.
Lemma 1. Given any two quantum channelsE :
B(HA) → B(HA0) and F : B(HB) → B(HB0), with unitary representations of a groupGdefined on all input and output spaces. Then we have the following:
1. Iso(E ⊗F)Iso(E)∧Iso(F)under the tensor product group action Ug(E ⊗ F) = Ug(E)⊗
Ug(F).
2. If B0 =A thenIso(E ◦ F)Iso(E)∧Iso(F).
3. If A = B and A0 = B0, and p some proba-bility 0 ≤p≤1, then Iso(pE+ (1−p)F)
Iso(E)∧Iso(F).
4. Iso(Ug◦ E ◦ Ug†) =g[Iso(E)]g−1.
These results also apply for quantum states, once we view the state as the state preparation map
1→ρ. We can now make precise a coarse-grained (necessary, but far from sufficient) requirement that a state σ allow the simulation of a channel E under symmetric dynamics and show that this
bound is tight in general. The proof is provided in Appendix A.
Theorem 1. If a system B in a state σB can be used to simulate a CPTP mapEunder symmetric dynamics, then Iso(σB) ≺ Iso(E). Moreover, if E has isotropy group Iso(E) then there exists a quantum system B and quantum state σB with Iso(σB) =Iso(E) that allows the simulation ofE.
This result gives the basic relation, in the coarse-grained picture, for when a quantum stateσBcan be used to simulate a quantum channel and also shows that this relation is a tight one. This is necessary in order for the isotropy subgroup ap-proach to be consistent with the parent resource theory, where one is concerned with the minimal resources needed to realise a particular quantum channel.
A similar result can be given in the case of ap-proximate simulation. For any target operation E we can define the noisy version given by
E= (1−) E+D (5)
whereDis the complete depolarization mapρ7→ 1
to an approximate, ‘isotropic’ simulation of the original map with noise parameter . Theorem1
extends to this case in an obvious way.
3.2 Is ‘shape’ a resource-theoretic property? While we normally associate measurable proper-ties with either projective measurements or more generally POVMs, there is an alternative way that is more general again. Specifically, a prop-erty is associated with a pre-order defined on the set of all quantum states. This recent ap-proach is called the resource-theory method, and has found success in areas such as entanglement, coherence, thermodynamics, and many other sce-narios [4,32–34]. Quantum maps that respect the pre-order are called ‘free operations’ and any real-valued function on quantum states that respects the pre-order is a measure of the property.
Given an ability to order the isotropy sub-groups of quantum states and channels under the group action, we can ask if a meaningful notion of ‘shape’ can be defined for quantum systems, along such a resource-theoretic line. This would essentially characterise the asymmetry of states and channels without reference to a measure.
For continuous Lie groups, we find that the or-bit of a state or channel is always a homogeneous space [35,36], specified by both the groupGand the particular isotropy subgroup H of the chan-nel. More precisely the orbit of the channel is M(E)and coincides with the quotientG/Hup to diffeomorphisms, which we write M(E) ∼= G/H (and likewise for states).1 This is what we call the ‘shape’ of a state or channel.
A simple example can be given for a single qubit state under anSU(2)symmetry, where the possible types of group orbit are
M
1
2[1+r·σ]
=
(
SU(2)/U(1)∼=S2 r6= 0 SU(2)/SU(2)∼={e} r= 0,
(6)
and these are illustrated in Fig. 3.
The lattice of isotropy subgroups gives us a way of comparing the ‘shapes’ of group orbits under a group actionG, withM(E)≺ M(F)iffIso(E)
Iso(F) (up to isomorphism of the isotropy sub-groups). Likewise for states, M(ρ) ≺ M(σ) iff
1
Technically we define the ‘shape’ of the group orbit with respect to the conjugacy class of the isotropy sub-groupH. This ‘shape’ is also called the Orbit-Type in the literature. See [37] for further details.
Figure 3: Basic orbits of states. Group orbits for a single qubit under the action of SU(2) shown in the Bloch sphere. There are only two possibilities in this case: a sphere for non-zero Bloch vectors (left), or a point for the symmetric maximally mixed state (right).
Iso(ρ) Iso(σ) (up to isomorphism). The con-vention here is to match with resource-theoretic measures of asymmetry [21, 23], so that one ‘shape’ is ‘bigger’ than another if it has a smaller isotropy group. For example, a single-qubit state with non-zero Bloch vector (M(ρ) ∼= S2) has a ‘bigger’ group orbit than the maximally mixed state (M(1/2)∼={e}). This gives valuable infor-mation regarding the resources required to per-form a quantum channel in the presence of sym-metry constraints.
The isotropy subgroup gives us a natural equiv-alence relation ∼ between states that behave in the same way under the group action. We can then say that ρ1 and ρ2 are related ρ1 ∼ ρ2 if we have that Iso(ρ1) = Iso(ρ2). This equiva-lence relation partitions the state space into sets C(H) :={ρ: Iso(ρ) = H}, and collects together all states that break the symmetry in the same manner. Taking the union ofC(H)corresponding to isomorphic H partitions the states and chan-nels according to their ‘shape’.
Viewed as a mapping, C maps from the sub-group lattice onto a partition of the state space, and so we can simply allow the partition to in-herit the lattice structure, and order subsets of states as C(H1)C(H2) wheneverH1≺H2.
Note however that the sets {C(H)} are not convex in general. For example a qubit system under G = SU(2), both |0i h0| and |1i h1| have the same U(1)isotropy subgroup, but 12(|0i h0|+
|1i h1|) = 121, which is symmetric. However, C(G) is always a convex set. This follows be-cause ifρ1, ρ2∈C(G)then Lemma1implies that GIso(p1ρ1+p2ρ2)Iso(ρ1)∧Iso(ρ2) = G, and so any mixture has the same isotropy subgroup.
∅ for all H0 H, since there are subgroups of SU(2) containing U(1) which do not appear as the isotropy subgroup of a single qubit state. Not all possible isotropy subgroups will be seen in a given system, and for a finite dimensional system many of the sets C(H) will be empty.
However in order to have a meaning-ful resource-theory interpretation, the coarse-grained ordering must be compatible with the set of free operations – namely the symmetric oper-ations. It is easy to show that this is in fact the case at the level individual quantum states.
Lemma 2. Under a symmetric operation E,
Iso(E(ρ))Iso(ρ).
This shows that the coarse-grained ordering of states is consistent with the set of free (symmet-ric) operations, as expected. However it does not imply that a functional mapping is defined on the sets C(H); it is possible to haveρ1 andρ2 in the same setC(H), but each get sent to different sets C(H10)andC(H20). This aspect means that inter-preting the coarse-graining as defining a quantum resource needs care.
3.3 The speck of dust argument – tiny pertur-bations to symmetric states & channels
Having outlined how the set of quantum states or the set of quantum channels of a quantum system is partitioned according to the symmetry action one might raise the following concern: what hap-pens if one perturbs a channel via some small perturbation? How does this affect its isotropy subgroup?
It is readily seen that one can make an arbitrar-ily small perturbation to any quantum channelE so as to break the symmetry in a “maximal” way. Informally this amounts to the “speck of dust” argument where, for example, a perfectly rota-tionally symmetric ball can be made completely asymmetric by sticking arbitrarily small pieces of dust onto it (Fig. 4).
In more abstract terms, one can use the princi-pal orbit-type theorem [37] for the space of quan-tum channels of a system to see that there is always a principal isotropy subset that forms a dense subset in the set of all quantum channels. Thus “most” quantum channels on a system will break a symmetry to a maximal degree (the prin-cipal isotropy).
Figure 4: The speck of dust argument. If we start with a perfectly rotationally symmetric object (left im-age) we can always add arbitrarily small perturbations that completely break the rotational symmetry (right image). However physically the fact that we have fi-nite resolution means that quantum states and channels should always be understood as being defined up to some –smoothing scale. This does not affect the resource-theoretic account and therefore the ‘shape’ is a robust feature in the presence of such tiny symmetry-breaking perturbations.
This argument of course applies equally well to the partitioning of the quantum state space into different subsets with fixed isotropies, however in both cases this does not pose a problem for our formalism for the following reasons. Firstly, in resource theories we are fundamentally interested in the most efficient use of resources and therefore our task is to look for those quantum resource states that break the symmetry not in a maximal way, but in a minimal way.
Secondly, at the level of channels, the most interesting and commonly addressed channels in quantum information science are not of princi-pal isotropy type — namely they have non-trivial isotropies. For example, consider quantum chan-nels on a single qubit, and the irreducible group action of G = SU(2). A typical quantum chan-nel on the qubit will only have a small resid-ual isotropy group Z2 = {1,−1} (the principal isotropy subgroup of the set of all qubit chan-nels). However consider the following standard single qubit channels:
1. The depolarizing channel: E(ρ) =pρ+ (1− p)(121), with0≤p≤1.
2. A rotation about the nˆ axis: Unˆ(θ) =
exp[−inˆ·σ].
3. A projective measurement along the nˆ axis: E(ρ) = P
4. Partial dephasing along the axis ˆn: E(ρ) =
pρ+(1−p)P
ktr(Πkρ)Πk, whereΠ0 =|nˆi hnˆ| andΠ1=1−Π0 =|−nˆi h−nˆ|.
5. A qubit state preparation channel: E(ρ) =
1
2(1+p nˆ·σ).
It is readily seen that the isotropy subgroup for channel (1) is SU(2), while for channels (2–5) it is aU(1)subgroup ofSU(2). The orbits will have ‘shapes’SU(2)/SU(2)∼={e} andSU(2)/U(1)∼=
S2respectively. None of these channels have prin-cipal isotropy type {1,−1}, however they are clearly key quantum operations in quantum in-formation science for which one might wish to determine the minimal quantum resources neces-sary to realise them.
Thirdly, the fact that something is “measure zero” in some space does not imply that it is physically irrelevant. For example, a 2-d plane is a measure zero set in three spatial dimensions, however this does not mean that anyonic physics or the quantum hall effect is irrelevant. Or of closer bearing on our present work, we can con-sider the case of Noether’s theorem: clearly the set of dynamics with rotational symmetry is again “measure zero”, however this does not mean that Noether’s theorem is irrelevant for physics. In both cases while the restrictions are strictly un-stable under small perturbations, the important thing is that the features of interest are opera-tionally robust under small perturbations.
The speck of dust subtlety makes itself appar-ent in our setting if one tries to naively exploit some metric on quantum states to quantify how far apart the sets {C(H)} are from each other. It is readily seen that the distance been any two such sets is in fact zero.
Lemma 3. Let d(·,·) be any metric on the space of quantum states. In terms of this metric we define
d(C(H1), C(H2)) := inf σ1∈C(H1)
σ2∈C(H2)
d(σ1, σ2). (7)
Then d(C(H1), C(H2)) = 0 for all H1, H2 ≺G. This also shows that any set C(H) is arbitrar-ily close to the set C(G). However, this simply means that any symmetry properties must always be considered up to somefinite resolution scale – arbitrarily small perturbations can always elimi-nate residual symmetries, even though this state
is practically indistinguishable from the unper-turbed one.
Therefore any membership of a quantum state ρ to a subset should only be considered up to some smoothing scalebased on a distance mea-sured(such as from theL1norm). For each state there is an -ball of nearby states, B(ρ) = {σ :
d(ρ, σ)≤}. Rather than Iso(ρ), we should con-sider Iso(ρ) := maxσ∈B(ρ)Iso(σ). Intuitively, if a state ρ is within a distance of another state with more residual symmetries, we asso-ciate those additional residual symmetries withρ. The smoothing scale selects the size of symmetry-breaking perturbations that we wish to consider, and will depend on the particular physical con-text.
This reasoning applies equally to quantum channels. For example, the depolarisation chan-nel on a qubit is fully invariant under the G =
SU(2) and so no non-trivial resource state is needed to simulate it. If one perturbs this channel to E → E by some >0perturbation in the dia-mond norm one can clearly break this isotropy to the principal isotropy {1,−1}, however it is also clear (e.g. from the continuity of the Stinespring dilation [38]) that E can either be simulated ex-actly with a resource state on B that -close to be being symmetric, or simulated approximately to the same threshold with a perfectly symmet-ric state on B. Thus the use of isotropy groups within the resource theory context is robust under such small perturbations and the speck of dust argument is moot. We illustrate this point ex-plicitly in Section 4, where we demonstrate that for the case of two qubits underSU(2)the notion of different ‘shapes’ is perfectly robust under the particular case of perturbations that allow a 4%
error rate.
3.4 Projecting out residual symmetries via quantum operations
Given the structure of states under the above partition, we can ask how easy it is to move from one set C(H) to another. As discussed one does not in general have quantum operations that map any C(H) neatly into some other sub-set. Instead it makes sense to consider the sets
ˆ
C(H) :=S
well-defined mapping of Cˆ(H)→Cˆ(f(H)). The quantum operation
PH(ρ) :=
Z
H
dhUh(ρ) (8)
is the average of the stateρover a fixed subgroup H weighted by the invariant Haar measure dh. In the case of a finite subgroup of size |H|, the integralR
Hdh is replaced by the sum 1 |H|
P
h∈H.
Lemma 4. The mapPH has the following prop-erties:
1. PH is the (orthogonal) projector ontoCˆ(H).
2. PH(ρ) = arg minσ∈C(H)ˆ S(ρ||σ), where S(ρ||σ) is the relative entropy.
Therefore PH projects onto Cˆ(H), as illus-trated in Fig 2. Dephasing [20], E(ρ) =
1 2π
R2π
0 dθ eiθZρe
−iθZ, can be viewed as P U(1), sending ρ to the nearest incoherent state while preserving the diagonal terms of a density ma-trix.
The mapping PH moves down chains of the subgroup Hasse diagram, however this is not al-ways a symmetric operation. The following tells us when such a transformation can be performed freely in the resource theory.
Lemma 5. Given a group action for G,
Iso(PH) = NG(H), where NG(H) = {g ∈ G : gHg−1 =H} is the normalizer [39] of H in G, and therefore if H is a normal subgroup of G (H / G) then PH is a symmetric operation.
This constrains the kind of resources needed to move from oneC(H)to another using PH, since this projects onto a given Cˆ(H), although less resource-hungry ways may exist to perform the same transformation.
3.5 Discussion of the section results
In this section we have given an analysis of the what happens when one classifies quantum states or channels in terms of their residual symmetries. We discussed how composition and mixing affect the ordering, and how one can relate these fea-tures to the issue of simulation (exact or approx-imate) of a target quantum operation with some resource state. We also saw that the ordering has subtleties if one wishes to interpret it in a re-source theoretic sense. The statements one makes
are also quite blunt, and a good example of this is the fact that the sets{C(H)}are all arbitrarily close to one another. However the relations still carry non-trivial content, and for example Theo-rem 1 can be viewed as a form of superselection rule on quantum operations that tells us which states are ruled out and which are not.
While this is conceptually neat and provides high-level insight into the complexities of the full classification of states (as described in [7]), it is less clear how computationally useful or simple these structures are in practice. To address this point, in the next section we consider the impor-tant case of a two-qubit system under an SU(2)
action. We find that already in just this sim-ple scenario the hierarchy of states is quite com-plex, however the example does provide insight into what to expect in the more general case.
4
Illustrative Example:
Two Qubits
Under
SU
(2)
Symmetry Constraints
The goal of this section is to illustrate the clas-sification of quantum states in a simple quan-tum system that has sufficient structure, yet is tractable to the point of being fully solvable. The state space of a two-qubit system is 15 di-mensional and is sufficiently non-trivial, more-over there is a very natural group action to con-sider, namely the tensor product representation of SU(2). The orbits of 2-qubit states have been studied in relation to thermodynamics and corre-lations within these states [40,41].
Any two qubit stateρAB can be written
ρAB =
1 4
1⊗1+a·σ⊗1+1⊗b·σ
+
3
X
i,j=1
Tij σi⊗σj
, (9)
with local Bloch vectors a andb and correlation terms determined by the correlation matrix Tij [42]. In general |a| ≤ 1 and |b| ≤ 1. Moreover, we can put all 2-qubit states into diagonal form,
ρAB =
1 4
1⊗1+a·σ⊗1+1⊗b·σ
+
3
X
i=1
τi ci·σ⊗di·σ
!
where {ci} and {di} are orthonormal bases of R3. The coefficients τ
i specify a posi-tion (τ1, τ2, τ3) in a tetrahedron with vertices
(−1,−1,−1),(1,1,−1),(1,−1,1) and (−1,1,1). When a = b = 0, then ρAB is a quantum state with maximally mixed marginals represented by a point in the tetrahedron. In the case a or b being non-zero, every quantum state ρAB corre-sponds to a point in the tetrahedron, however the converse is not true: not all triples (a,b,t) cor-respond to a valid quantum states.
Via local unitariesU1⊗U2, these can be trans-formed into the canonical form
˜
ρAB =
1 4
1⊗1+a0·σ⊗1+1⊗b0·σ
+
3
X
i=1
τi σi⊗σi
!
(11)
wherea0·σ=U1(a·σ)U1†andb0·σ =U2(b·σ)U2†. The states in this canonical form with maximally mixed marginals are called T-states [42],
ρT =
1 4
1⊗1+ X
j
τj σj⊗σj
. (12)
The set of T-states is the convex hull of the four Bell states. The corners of the tetrahedron de-fined by the coefficients {τi} correspond to the Bell states.
We now consider the symmetry properties of 2-qubit states under SU(2) symmetry constraints, assuming an SU(2) tensor product representa-tion for the group acrepresenta-tion, i.e. Ug(ρ) = U(g)⊗ U(g) ρ U†(g)⊗U†(g), where U(g) is a two di-mensional irrep ofSU(2).
Our description of the group orbit as the quo-tient space G/Iso(ρ) starts with the group man-ifold of Gand eliminates the redundancy caused by the symmetries of ρ. The group transfor-mations in Iso(ρ) have trivial group action on ρ, so we quotient out the equivalence classes g Iso(ρ) :={ghi: hi ∈Iso(ρ)}.
Certain groups have structure which permits us to further simplify our description of the group orbit manifold. Here we consider SU(2) symme-try constraints, however thisSU(2) action has a subgroup Z2 ={±1} that is always a symmetry of a quantum state, and which is associated with SU(2) being the double cover of SO(3). More-over this sub-group has a key property which al-lows us to simplify our description of the SU(2)
isotropy groups to those ofSO(3). We can there-fore think in terms ofSO(3)subgroups, which are the familiar chiral point groups one encounters in for example crystallography [1].
A subgroup N of G is a normal subgroup, de-noted N / G, if it satisfies gN g−1 = N for all g ∈ G [1]. These partition G into equivalence classes gN :={gni : ni ∈N}, which themselves constitute elements of the quotient group G/N. Every element of G is specified by the pairing of the equivalence class gN and an element of the normal subgroupN. Likewise, elements ofH are specified by the pairing of an element ofH/N and an element ofN becauseN / H. This means that the group orbit manifold G/H can be described by the quotient groups G/N and H/N, and we can write M(ρ) ∼= G/H ∼= (G/N)/(H/N). In our present analysis Z2 is a normal sub-group of SU(2) common to all quantum states and so M(ρ) ∼=SO(3)/(H/Z2) where H is the isotropy group of ρ in SU(2). We can therefore use the more intuitive SO(3) subgroups, listed in Table
1.
Group Description
Cn Cyclic group of ordern
C∞ Symmetries of Cone
Dn Symmetries of a regularn-sided Polygon D∞ Symmetries of Cylinder
T Symmetries of Tetrahedron
O Symmetries of Cube/Octahedron
I Symmetries of Dodecahedron/Icosahedron
Table 1: Chiral point groups: subgroups of SO(3),
as detailed in [1]. All quantum states have isotropy sub-groups under SU(2) that can be related to particular chiral point group.
In the next section we look at the isotropy sub-groups of the Bell states, and then go on to classify the symmetry properties of the T-states. Then we continue onto the wider class of 2-qubit states with maximally mixed marginals and fi-nally we complete the classification for general 2-qubit states by considering non-zero local Bloch vectors.
4.1 Bell States
calculated directly, as described in Appendix B. The singlet Bell state ψ− is symmetric un-der SU(2), therefore Iso(ψ−) = SU(2) and so M(ψ−) = {e} as expected. The triplet Bell states do break the SU(2)symmetry, with
Iso(φ+) ={(iZ)αeiθY : 0≤θ <2π, α= 0,1,2,3}.
(13)
We will call this subgroupK∞∼=U(1)oZ4, and it provides an example of a semi-direct product group2. Similarly,
Iso(φ−) ={(iY)αeiθX: 0≤θ <2π, α= 0,1,2,3}
(14)
and
Iso(ψ+) ={(iX)αeiθZ: 0≤θ <2π, α= 0,1,2,3},
(15)
both of which are isomorphic to K∞.
Given the triplet Bell states have isomorphic isotropy subgroups, their group orbits will have the same shape, M(φ+) ∼=M(φ−) ∼=M(ψ+) ∼=
SU(2)/K∞. In the previous section we described how normal subgroups simplify our description of the group orbit, and we can use this here to put the description in terms of SO(3) sub-groups. There is a 2-to-1 homomorphism from the quotient G/{1,−1}. Elimination of the Z2 normal subgroup from the group orbit descrip-tion gives SU(2)/K∞ ∼=SO(3)/D∞, where D∞ is the isotropy subgroup of the cylinder.
4.2 T-States
From the Bell states, we can begin to map out the possible shapes of group orbit in the tetrahedron
2 The semi-direct product [1] is constructed from two groups, in this case Z4 and U(1). Each element of the
semidirect product group can be thought of as a pairing (h, n) whereh∈H =Z4 andn∈N=U(1). However the
multiplication law of the semidirect product group does not treat the elements of this pairing independently, with (h1, n1)(h2, n2) = (h1h2, fh2(n1)n2) wherefh : N → N
is a automorphism on the second group specified by an element of the first group. The semi-direct product o
is defined by the choice of automorphism fh. Note that
N /(NoH). If a trivial (identity) automorphism is
cho-sen, where fh(n) = n for any hand n, we have a direct
product group [1]. In the U(1)oZ4 example, the group
multiplication law is
(iZ)α1eiθ1Y
(iZ)α2eiθ2Y
= (iZ)α1+α2ei[(−1)α2θ1+θ2]Y
.
of T-states. Consider the convex hull of triplet Bell states to be the base of the tetrahedron, and ψ−to be the peak. The vertical height above the base of this tetrahedron indicates the proportion of the singlet state in the convex mixture p1φ++ p2φ−+p3ψ++p4ψ−, wherep1+p2+p3+p4 = 1. Sinceψ− is symmetric under the SU(2)group action,
M(p1φ++p2φ−+p3ψ++p4ψ−)
=M(p1φ++p2φ−+p3ψ+), (16)
therefore the possible types of group orbit for T-states will all be exhibited in the convex hull of the triplet states. These can be visualized as a triangle of states, as shown in Fig. 5.
The vertices of this triangle are the triplet Bell states, and each have a K∞ isotropy subgroup. Appendix B also shows that the midpoints of the edges of this triangle, corresponding to states
1
4(1⊗1+σi⊗σi), also have K∞ isotropy sub-groups. Therefore the diagonals of this triangle, corresponding to states
ρ=pφ++ (1−p)1
4(1⊗1+Y ⊗Y) (17)
ρ=pφ−+ (1−p)1
4(1⊗1+X⊗X) (18)
ρ=pψ++ (1−p)1
4(1⊗1+Z⊗Z) (19)
for 0 ≤ p < 1, have K∞ ≺ Iso(ρ). The states not on these diagonals remain invariant under the subgroup K2 ={±1,±iX,±iY,±iZ}.
The possible isotropy subgroups exhibited by the T-states can be seen with the projectors PH, as detailed in AppendixC. These are:
• Iso(ρT) = SU(2)⇒ M(ρ)∼={e}
Werner states along the central axis of the T-state tetrahedron, where τ1 = τ2 = τ3. These are the states
ρ= (1−p) ψ−+p 3 (ψ
++φ++φ−
) (20)
for 0≤p≤1.
• Iso(ρT) = K∞⇒ M(ρ)∼= SO(3)/D∞
Figure 5: The partitioning of the2-qubit state space. a) Tetrahedron of T-states coloured to indicate different group orbit shapes: grey – SU(2)/K2 ∼= SO(3)/D2, light blue – SO(3)/D∞, black – {e}. Slices through the tetrahedron that are parallel to the plane containing the triplet Bell states exhibit the same structure, exhibited in the triangle to the right. b) Tetrahedron mapping out the smoothed isotropy subgroups with smoothing scale = 0.04. The bands along the diagonals indicateIso(ρ)∼= K∞, while states in the central hexagon are assigned Iso(ρ)∼= SU(2). The remaining states retain theirK2isotropy subgroup. c) Hasse diagram of the observed isotropy
subgroups (up to isomorphism) for two qubits. Arrows indicate subset inclusion. This is a sublattice of the fullSU(2) subgroup lattice.
• Iso(ρT) = K2 ⇒ M(ρ)∼= SO(3)/D2
T-states withτ1 6=τ26=τ3. These are states that do not lie on or above the diagonals of the triangle formed by the triplet Bell states.
The smoothed isotropy subgroups are illustrated in the lower part of Fig. 5, with a smoothing scale of= 0.04under the trace distance. Under this smoothing, the horizontal slices through the tetrahedron are no longer equivalent; near ψ−all states are assigned SU(2) as their isotropy sub-group, while at the base there remain many states withIso(ρ) = K2.
4.3 Maximally Mixed Marginals
The set of 2-qubit states with maximally mixed marginals is a wider class of states reached from the T-states through local SU(2) unitaries U(g1) ⊗ U(g2). Appendix C details how the isotropy subgroups are enumerated.
If these local unitaries are the same for each qubit, we reach states of the form
˜
ρT =U(g)⊗U(g) ρT U†(g)⊗U†(g) =Ug(ρT)
= 1
4 1⊗1+
3
X
i=1
τi ci·σ⊗ci·σ
!
, (21)
where the vectors{ci}form an orthonormal basis of R3.
These states behave similarly to the T-states, but with isomorphic isotropy subgroups. This can be seen from the final part of Lemma1, where Iso(Ug ◦ρT ◦ Ug−1) = g[Iso(ρT)]g−1. Explicitly,
instead of {±1,±iX,±iY,±iZ}, these rotated T-states will be invariant under {±1, ±i(c1 · σ), ±i(c2·σ), ±i(c3·σ)} ∼=K2.
The possible types of group orbit for these states are the same as for the T-states:
• Iso( ˜ρT) = SU(2)⇒ M( ˜ρT)∼={e}
U(g) ⊗ U(g). These are states ρ˜T with τ1 =τ2 =τ3.
• Iso( ˜ρT) = K∞⇒ M( ˜ρT)∼= SO(3)/D∞
States ρ˜T with τi = τj 6= τk. These are reached via unitary transformations U(g)⊗ U(g) from the T-states with K∞ isotropy subgroups.
• Iso( ˜ρT) = K2 ⇒ M( ˜ρT)∼= SO(3)/D2
States ρ˜T with τ1 =6 τ2 6= τ3. These are reached via unitary transformations U(g)⊗ U(g)from the T-states withK2isotropy sub-groups.
Further states with maximally mixed marginals are reached from the T-states by different local unitaries on each qubit. Suppose that these two local unitaries have the same generatorr·σ, such thatU(g1)⊗U(g2) =eiθ1r·σ⊗eiθ2r·σ. This gives states of the form
ρM =eiθ1r·σ⊗eiθ2r·σ ρT e−iθ1r·σ⊗e−iθ2r·σ
= 1
4 1⊗1+τ1 r·σ⊗r·σ+
3
X
i=2
τi ci·σ⊗di·σ
!
(22)
where {r,c2,c3} and {r,d2,d3} are two sets of orthonormal bases forR3. IfU(g1) =U(g2)in the local unitaries, these sets of bases will be the same and we retrieve ρ˜T. However, in the case that θ1 6= θ2 in the local unitaries, new possibilities for the isotropy subgroup are introduced:
• Iso(ρM) = U(1)⇒ M(ρM)∼= S2
States ρM with τ2 =τ3. These are reached from T-states with K∞ isotropy subgroups via local unitarieseiθ1r·σ⊗eiθ2r·σ whereθ
1 6= θ2.
• Iso(ρM) =Z4⇒ M(ρM)∼= SO(3)/C2
States ρM with τ2 6=τ3. These are reached from T-states with K2 isotropy subgroups via local unitarieseiθ1r·σ⊗eiθ2r·σ whereθ
1 6= θ2.
The statesρ˜T are only invariant under a K2 sub-group because the correlation terms are put into diagonal form by the same orthonormal bases {ci} and {di} on each qubit. For example, the correlation terms of the T-states are put into the canonical form by the standard cartesian or-thonormal basis {x,ˆ y,ˆ ˆz} on each qubit. When
the local unitaries are applied, this can break the K2 symmetry, and the local bases become ‘mis-aligned’. However, when c1 = d1 = r there is still a subgroup Z4 ={±1,±i(r·σ)} that leaves these states invariant.
The remaining 2-qubit states with maximally mixed marginals are reached from the T-states through local SU(2) unitaries that do not share a generator, such thatU(g1)⊗U(g2) =eiθ1r1·σ⊗ eiθ2r2·σ where |r
1|=|r2|= 1 andr1 6=r2. These states can be expressed in the form
˜
ρM =eiθ1r1·σ⊗eiθ2r2·σ ρT e−iθ1r1·σ⊗e−iθ2r2·σ
= 1
4 1⊗1+
3
X
i=1
τi ci·σ⊗di·σ
!
, (23)
where{ci}and{di}are orthonormal bases ofR3 that do not share any elements, i.e. ci 6=dj for all i and j. These states exhibit another type of group orbit:
• Iso(ρ) =Z2⇒ M(ρ)∼= SO(3)
States ρ˜M. The orthonormal bases{ci} and {di}do not share any elements, therefore the only subgroup that leaves these states invari-ant isZ2 ={±1}. These are in a sense max-imally asymmetric states, as they have no non-trivial residual symmetries.
4.4 Local Bloch Vectors
Finally, the introduction of local Bloch vectors completes the set of 2-qubit states. In the states with maximally mixed marginals, the alignment of the orthonormal bases describing the corre-lation terms played a key role in determining the symmetry properties. The effect of the lo-cal Bloch vectors depends on the how they relate to each other, as well as how they relate to the bases for the correlation terms.
There are three possible isotropy subgroups for these states:
• Iso(ρ) =Z2⇒ M(ρ)∼= SO(3)
This will occur for any states
ρ= 1 4
1⊗1+a·σ⊗1+1⊗b·σ
+
3
X
i=1
τi ci·σ⊗di·σ
!
whose local Bloch vectorsa·σ and b·σ are not parallel or anti-parallel, such that a 6=
γb for any γ ∈R.
This isotropy subgroup will also occur in states where the basis in which the correla-tion terms are diagonalised does not align with the local Bloch vectors. These are states of the form
ρ= 1 4
1⊗1+a·σ⊗1+1⊗b·σ
+τ1 r·σ⊗r·σ+ 3
X
i=2
τi ci·σ⊗di·σ
!
(25)
where{r,c2,c3}and{r,d2,d3}are two sets of orthonormal bases for R3 and either a or b 6= γr for any γ ∈ R. In this case, the correlation terms would be invariant un-der Z4 = {±1,±ir · σ}, however the lo-cal Bloch vectors prevent this. Similarly if τ2 = τ3 in states of this form, then the lo-cal Bloch vectors break a U(1) symmetry, while if {ci} = {di} then the local Bloch vectors break a K2 symmetry. Likewise if both {ci} = {di} and τ2 = τ3, the corre-lation terms are invariant under a K∞ sub-group that is again broken by the local Bloch vectors.
• Iso(ρ) =Z4 ⇒ M(ρ)∼= SO(3)/C2
States of the form
ρ= 1 4
1⊗1+ar·σ⊗1+b1⊗r·σ
+τ1 r·σ⊗r·σ+ 3
X
i=2
τi ci·σ⊗di·σ
!
(26)
where{r,c2,c3}and{r,d2,d3}are two sets of orthonormal bases for R3, a, b ∈ R and τ2 6=τ3. The local Bloch vectors are invari-ant under the sameZ4 subgroup that leaves the correlation terms invariant. This also in-cludes the case where ci = di for i = 2,3. The correlation terms are invariant under a K2 subgroup, but the local Bloch vectors break this to aZ4 subgroup.
Similarly, states of the form
ρ= 1 4
1⊗1+ac1·σ⊗1+b1⊗c1·σ
+
3
X
i=1
τi ci·σ⊗ci·σ
!
(27)
where{c1,c2,c3}is an orthonormal basis for R3 and τ1 =τ2 6=τ3. The correlation terms are invariant under aK∞ subgroup, but the local Bloch vectors break the residual sym-metry down to a Z4 subgroup as the local Bloch vectors are orthogonal to c3 ·σ, the generator of the relevantU(1) subgroup.
• Iso(ρ) = U(1)⇒ M(ρ)∼= S2
These are states
ρ= 1 4
1⊗1+ar·σ⊗1+b1⊗r·σ
τ1r·σ⊗r·σ+τ 3
X
i=2
ci·σ⊗di·σ
!
(28)
where{r,c2,c3}and{r,d2,d3}are two sets of orthonormal bases for R3, a, b ∈ R. The local Bloch vectors are invariant under the same U(1)subgroup that leaves the correla-tion terms invariant. This also includes the cases when {ci} = {di} and when τ1 = τ, where the correlation terms are invariant un-derK∞ and SU(2) respectively.
This completes the classification of the isotropy subgroups of 2-qubit states.
4.5 Some simple consequences of the classifi-cation
This classification allows us to infer constraints on the resources required to simulate asymmetric channels under symmetry constraints. If we wish to simulate an axial channel (i.e. M(E) = S2) under SU(2) symmetry constraints using the T-states, the resource state must have M(σ)S2. T-states on the diagonals of the tetrahedron, with M(ρ) ≺ SO(3)/D∞, will not be able to per-form such simulations; only the T-states with τ1 6= τ2 6= τ3 will achieve this task up to some approximation as discussed earlier.
(1−p)φ−) = K2whenp= 16 /2, despiteIso(φ+)∼= Iso(φ−)∼= K∞.
Group orbits also constrain the dynamics of the T-states. For instance, Lemma2implies that symmetric operations move states situated on a diagonal of the tetrahedron only along that diag-onal. The T-states are closed under symmetric dynamics – leaving would entail further symme-tries being broken.
Also note that H1 ⊆ H2 does not imply there exists a symmetric channel fromC(H1)into C(H2). T-states on the diagonals of the base of the tetrahedron cannot be reached from those that lie off these diagonals under symmetric op-erations [25], despite K2 ≺ K∞. Group orbits impose the highest level constraints; further tech-niques such as those of [25] then impose more de-tailed constraints.
5
Outlook
Our analysis has shown that the basic prob-lem of state interconversion under a symmetry constraint has a rich and non-trivial structure. We have established a high-level description of the problem that is consistent with the resource-theoretic picture, however the question whether the framework can find practical use in the same way that superselection rules simplify computa-tion remains to be explored.
We have classified the set of all two-qubit states under the SU(2) tensor product representation, which may be of use in its own right for the study of quantum correlations. However, beyond two-qubits, the general problem can be extremely dif-ficult – for example even the finite subgroups of SU(5)remain unclassified [17]. Despite this com-plexity, we believe that the tools we have devel-oped here provide useful insight into the struc-ture of information processing under symmetry constraints.
Other techniques for studying the effect of sym-metry constraints on quantum operations con-nect with the present analysis. As already men-tioned, recent work on the harmonic analysis of quantum channels suggests that the geometry of group orbits provides an insight into the use of finite quantum resources [25]. The present paper has focused purely on the ‘shape’ of this orbit, but the role of the induced geometry on the or-bits is only partially understood. To extend such
an analysis would require adapting techniques in metrology [15] and differential geometry [43]. It would also be of interest to draw connections with recent work [44] that extends classical coarse-graining into the quantum regime with symme-tries present.
Finally, these ideas may inform us about other resource theories, using the symmetry constraints directly (e.g. in thermodynamics). Along similar lines, the techniques we have described may find application in the context of realising universal quantum computation via the combination of re-source states with simple gate-sets, for example Clifford operations and magic states.
6
Acknowledgements
We would like to thank Cristina Cirstoiu, Erick Hinds Mingo, Zoe Holmes and Markus Frembs for many useful discussions. TH is funded by the EPSRC Centre for Doctoral Training in Con-trolled Quantum Dynamics. DJ is supported by the Royal Society.
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